Description Logic in 12 Minutes

DL is a decidable fragment of FOL

Description Logic gives you most of first-order logic's modelling power while keeping reasoning decidable (an algorithm always terminates). The trade is intentional: less expressivity, predictable performance.

Three primitives

• Concept (≈ OWL class) — a unary predicate. Person, Doctor.
• Role (≈ OWL object property) — a binary predicate. hasChild, worksFor.
• Individual (≈ OWL named individual) — a constant. :alice.

The ALC operators (the alphabet that gave its name to the family)

Symbol	Reading	OWL counterpart
⊓	intersection	owl:intersectionOf
⊔	union	owl:unionOf
¬	complement	owl:complementOf
∃ R.C	exists R into C	owl:someValuesFrom
∀ R.C	for all R into C	owl:allValuesFrom
⊑	subsumption	rdfs:subClassOf
≡	equivalence	owl:equivalentClass

Example: a HappyParent is a parent all of whose children are happy.

$$HappyParent \equiv Parent \sqcap \forall hasChild.Happy$$

Think of DL as Lego with shape rules:

• Each brick is a concept. Each connector is a role.
• The rules (⊓, ⊔, ∃, ∀) tell you how bricks may snap together.
• The reasoner is the instruction booklet: from the bricks on the table it derives every legal sub-assembly you didn't explicitly build.

Crucially, the rules are designed so the booklet always terminates. That's decidability — and it's why DL exists at all rather than just using FOL.

Tools & resources

• DL Handbook (Baader, Calvanese, McGuinness et al.) — the canonical reference: https://www.cambridge.org/core/books/description-logic-handbook/
• OWL 2 Primer — W3C, the gentlest official intro: https://www.w3.org/TR/owl2-primer/
• Protégé + Pizza Ontology — the world's most-used hands-on DL tutorial: https://protege.stanford.edu/ontologies/pizza/pizza.owl
• Description Logic Complexity Navigator — see exactly what each DL feature costs: http://www.cs.man.ac.uk/~ezolin/dl/